Post on 25-Jan-2021
Lecture 19 Geometricmodularforms
Dec 21Theorem X N GINN parametrizesall elliptic curves over togetherwith
an N torsionpoint
Explicitly for 2Efg Ito Zz f CEeProof Note if Ezi E q zz Ez E 2 2z
themapmustbegivenbymultiplicationbysome WEEthen 1 1 CtdZ w c dz E
Z't a be ctdz
Yet Tv fumod 2 7Lzi e F mod I Izadz 1 mud NI NIE1 D o mod N D
Theorem Yoln b ToCN parametrizesallellipticcurves over1CtogetherwithacyclicsubgpoforderN Explicitly for 2 ctry Ee 6 2 02 z 742Eez
Fact Wehavea universalfamilyofellipticcurvesE
YEEsilly.cn is locallyfreeofrank1 over E
I jezewsutin w t Ryy EEREH.cnY.CN bhHn 2 hasthedifferentialsheafon Etisthetrivialsheaf
w is a linebundle on Yi N
Geometricinterpretationf modularformsVersion 1 Ez OneachEz there's a canonicaldifferentialfan de
ferenhalfamdEt 7 2e
Y N As we usetheisomorphism Ez EzT Ii w adz c et
For amodularformfofweightb levelTIN we defineEz 1 1 ftz de kItsEei sifczgxodeixok_tlEEIdbtdIjIfczsdexokS.oL Sectionof REGI.hu is equivariantforT.CN action
Z 1 of DEOK
fxodiok fetffy.CN k
Version2 whenhis even note fC dzkkonly is invariantunderh N actionIndeed ffafzt.IT dfaIIbYhk Ccztdshfczs.f jdtbIDoI Ih f d k
So a modularform fmaybeviewedasaseetinfczsdEHZEHYYCN.rf.cn kBmt Kodaira Spencerisomorphism w 2 Ryfµy
Extensionto cusps X ftp.yfnlufcusps b nmuPkQYifn11
puniversalellipticaureC
Then E YE ZE sit over cusps wehave generalizedellipticcurvesat Je t Je for 2EC Ez lookslike or
UIYin XinEsg Gm or Em eafstairrepdi.com
Esmis a groupvariety overXelN
gray elywon YilN extends to wi erfsnyx.cn
Kodaira Spencerisomorphism w2 Nµy logcusp
differentialsheafwithlogpoleatthecusp
i e if 9 isthelocalparametuatacuspafflog has localbasisdftasopposedtdg
As Xin isawweidx.qylbg4 rx.cn ink
Geometricinterpretation Mahim H X N wh
Ul vanishingatcuspsshkilND HYX.CN w hfd
whenk z KS Isom MATCH HYXi N R'xWkycUl
szkilND HOCX.CN asHeckeoperators SayptN if cyclicsubgpoforderp
XftolpN Z Ez.subgptv74z.EE
x It I tX N X N Z Pt Estate Ek Cutch
9Version1 Consider E Ez isogenyI
E e
XIN
OnXoCpN we havetwolinebundles Tfw wza naturalmap wz q w
1naive youwww.IEEHYxcni wxok 9 isHYxCn wxok
Ip H XIN W I H XolpN bWz DH XopN h W
T H x n wit
Define tp p pai.sk ToND aSkToCNDExercise Verifythatthis is thesame astheearlierdefinitionRmtForNN level weusethesamesetup butthemap it is a littledifficult
towrite in termsof zh anettofECNofexaetorderNVersion2 whenh 2 E PnCptfEYffff
IT TE order
p.aeX CERT ftp.pntcpkpRealizing f Tim HYX.CN as
ThenTp HYX.CN.sh.cn E sHYxfNnhtoCpD.N E tifx.ln Nx.im
lls
tilxim.ci2 E.tifxaiiniiiocpi.wEgEIoEtiCx.iii.wioy
Forweightk z we alsohavethefollowingconstruction
ftp.tifx.lnl.sh.cnv
tifxfinhtocpD.ITIsHofx.Cns.Nx.cmYHifxIniiajE.tHxa
intropisia thx said
ftp.Y J.CN JATIN nTolp J.CNUpshot
Softy Jj abelianvarietywithHeckeations
HodgeTheoryForanysmoothproj came X la H x 2 CI HIX HKxSo H x 2 Cl thx HKX.IT
In particular H X N 2 a SktCNN SKITCorollary EigenvaluesofTp are algebraicintegers whenh
This is because Tpv H x N 2 th x N 2
hascharpolynomial in 21 1Cansimilarlydefine forplN Up HYX.CN in i HYX.CN ng
X NN nToGoND E 9 E
Y ti I tX 1N Xi N E E
na J N J N
f de 74Na Sd X N X N
EP B E d P
na Ld J N Ji NMoregenerally considerthenaturalmap
If II I End Jim End Hain acommutativealgebra finiterankover 2
Let h.IN imagefthismap whichisfinite12
TheoremThere is a perfectpairing h Nic x SKIN Eh fl a a Chefs
Proof nondegeneracy at Sa If fesdr.cm sit thekiln a chefsYetforh Tu a TnLlp an f o f o
f f f fHereTn is def'dby Tnm Tn Tm whengcdcn.in 1
Tph TpTph i XCpl.ph phzifptNTpkUphifpNnondegeneraeyathilNa If Teh.CN a is suchthat a Tf o tf
Then a Tnf a TnTfl anTfTf off 7 o D
Corollary H X Maio Cl Salman SachN1T u v
h.lnlxoqflatofrkzh.CN xoCh.ln xoEfreeofrk t feeofranktfinited.milArtinian
So HfXdn5Q isfeeofrankzoverh.CN QRmt Theintegralversionmightbebad
normalized eigencuspfoms f Eangf in SCN X
Ibijection Ihomomorphisms tfftp.fapptNdfih.IN sQlf 1C HCUp applN
f HkdXCd def74ve
eigensparequotientsof J.IN LetIf kerlhfihilN ilQlfi 1
Salt D tfDefineAg JdNYIfIµ H xcnn.tl IfH
Then Ag is an abelianvariety over ofdim IQlfi.IQTheGala'srep'sassociated tof is VelAf XiyuRemark f is always atotallyrealoracMfieldm
f ay lolally fat'otallginaginagguadratriextaofatotally
realfield